The Illini method for simplifying a radical


One of my linear algebra students is an education major doing student teaching. Today he showed me this method of simplifying radicals which he learned from his supervising teacher. Apparently it’s called the “Illini method”. Googling this term returns nothing math-related, so I think that term was probably invented by his supervisor, who went to college in Illinois.

The procedure goes as follows. Start with a radical to simplify, say \sqrt{50}. Look under the radical and find a prime that divides it, say 5. Then form a two-column array with the original radical in the top-left, the divisor prime in the adjacent row in the right column, and the result you get from dividing the radicand by that prime number in the left column below the radical. In this case, it’s:

\begin{array}{r|r} \sqrt{50} & 5 \\ 10 &  \end{array}

Now look for a prime that divides the lower-left term, say another 5. Again, put the dividing prime across from the dividend, and the quotient below the dividend. With our example, the array at this stage looks like:

\begin{array}{r|r} \sqrt{50} & 5 \\ 10 & 5 \\ 2 &  \end{array}

In general, continue this process of dividing prime numbers into the lower-left entry in the array, writing the prime across from that entry, and writing the quotient beneath that entry, until you end up with a 1 in the lower-left entry. So the final state of our example would be:

\begin{array}{r|r} \sqrt{50} & 5 \\ 10 & 5 \\ 2 & 2 \\ 1 &  \end{array}

Now, look at the left-hand column of the array. Group off any pairs of numbers you see. Multiply together all numbers which are representative of a pair. In our case, there is only one such pair, a pair of 5’s. Any numbers that occur singly are placed under a radical and multiplied. In our case, that’s the single 2. Then multiply the product of numbers which are in pairs times the radical which contains the singleton numbers. So we end up in our example with 5 \sqrt{2}.

Here’s another example with a larger number, \sqrt{2112}:

\begin{array}{r|r} \sqrt{2112} & 2 \\ 1056 & 2  \\ 528 & 2 \\ 264 & 2 \\ 132 & 2 \\ 66 & 2 \\ 33 & 3 \\ 11 & 11 \\ 1 & \end{array}

There are three groups of 2’s, so outside the final radical we’ll put 2 \cdot 2 \cdot 2 = 8. And the 3 and 11 are by themselves, so under the radical we put 33. Hence \sqrt{2112} = 8 \sqrt{33}.

Pretty clearly, all this method is doing is presenting a different way to do the bookkeeping for doing the prime factorization of the number under the radical. The final step of grouping off the prime pairs and leaving the un-paired primes under the radical is analogous to finding all the squared primes in the prime factorization.

This method is nice and systematic, and I can see why students (and student-teachers) might like it. But it seems to be obscuring some important concepts that students ought to know. With the method of factoring, looking for squared primes, and then removing them from the square root, at least you are dealing directly with the inverse relationship between squares and square roots. The Illini method, on the other hand, uses an approach of “put this here and then put that over there” with minimal contact with actual math. It does work, and it does keep things in order. But do students really understand why it works?

Your thoughts?  What does this method make clearer, and what does it obscure? Should high school algebra teachers be teaching it?

10 Comments

Filed under Education, High school, Math, Teaching

10 responses to “The Illini method for simplifying a radical

  1. I really like that it gives them a more ordered result than the tree method that they use in middle school (a vertical list instead of a bunch of randomly spaced branches) and I also like that it makes them think “the square root of 2 squared is 2”, because I have had students who could tell me that the square root of 4 was 2, but couldn’t tell me that the square root of “2 squared” was 2.
    On the downside, I always *really hate* giving them steps to memorize & follow, because steps are so easily forgotten where understanding, though it takes longer to achieve, lasts longer. And I agree with you that some students would use the steps without really understanding what a square root actually is.
    I think if I were going to use this, I would have a day of doing numbers small enough that they can be factored mentally (like 12 and 45) so that they really understand why we write the “product of all numbers that are representative of a pair.” Then I would introduce the larger numbers on the next day and give them this method as a tool to find the factors easily.

  2. I share your reservations.

    I modify the traditional approach slightly to include a “separation step” (I teach generally strong high school and generally weak college students) There is nothing amazing, but it cuts down on the number of kids who try to simplify \sqrt{75} and get 3\sqrt{5}.

    Let’s see if I can handle the LaTeX:

    First, responsibility for locating perfect squares is theirs.

    \sqrt{252x^5y^2}
    \sqrt{36x^4y^2\cdot 7x}
    \sqrt{36x^4y^2} \sqrt{7x}
    6x^2|y| \sqrt{7x}

  3. Jason Roy

    I have been really enjoying this blog for the past couple of weeks. I generally teach my students to simplify radicals by thinking about the largest perfect square that goes into the radical and then moving on from there. So Sqrt(80) = Sqrt(16*5) = 4Sqrt(5). It is sufficient for the problems with radcials students are likely to encounter that they would have to tackle without a calculator. I can’t think of too many situations when one would have to take the square root of 2112 by hand.
    I like to think that I get my students to understand as many concepts as possible but at the same time there are occasions when I just have them memorize things or use a device. This “Illini” method seems fine if the teacher takes the time to explain to the students why it works the way it does. It seems similar to the strategies I might use to change a number from base 10 to a different one and also to the Euclidean Algorithm.
    Finally I think there are countless situations where teachers do teach students a concept and then a clever algorithm to apply the concept and what happens is students proceed to remember the algorithm and forget the concept.

  4. Ben Chun

    I think the best use of this method would be after students already understand that:

    1. the square root of x squared is x (for non-negative x)
    2. the square root of a number is the same as the product of the square roots of its factors

    AND can demonstrate that understanding by solving a variety of problems such as the one you show in your comment. Only then should they see this method. After seeing the technique and using it a few times, the assignment should be to explain why it works — not to crank through pages of problems. The writing and thinking they do could be of much greater value than repeated applications of the algorithm. For struggling students I might suggest that they redo the problem using the traditional method to see if they can make some connections to write about. Stronger students should be pushed to explain things as accurately and precisely as possible, using mathematical language, and to build examples using variables. This is an activity that is inherently differentiated. (Feel free to put that in the essay you have to turn in for your credential program or masters degree or whatever.)

    This technique represents a great learning opportunity. But no one should be teaching it as a method for simplifying square roots before students are facile with the two concepts listed above!

  5. I think the best use of this method would be after students already understand that:

    1. the square root of x squared is x (for non-negative x)
    2. the square root of a number is the same as the product of the square roots of its factors

    AND can demonstrate that understanding by solving a variety of problems such as the one you show in your comment. Only then should they see this method. After seeing the technique and using it a few times, the assignment should be to explain why it works — not to crank through pages of problems. The writing and thinking they do could be of much greater value than repeated applications of the algorithm. For struggling students I might suggest that they redo the problem using the traditional method to see if they can make some connections to write about. Stronger students should be pushed to explain things as accurately and precisely as possible, using mathematical language, and to build examples using variables. This is an activity that is inherently differentiated. (Feel free to put that in the essay you have to turn in for your credential program or masters degree or whatever.)

    This technique represents a great learning opportunity. But no one should be teaching it as a method for simplifying square roots before students are facile with the two concepts listed above!

  6. All this really seems to do is turn what one might normally write horizontally into a vertical process. I don’t see how that’s an improvement, plus it takes understanding anyway to scale to cube roots and higher.

  7. I think cube roots are easy — the only change to the algorithm is to group off triples of primes in the right column instead of pairs.

    I still worry, though, that students don’t realize that what’s really happening in the right column is multiplication, not just a sort of vertical stacking and grouping.

  8. I would agree with Ben Chun. The method is interesting and it can be used to build the understanding of algebra students. The idea about having students explain why the method works is a great one. I also share his fear that if this method is introduced too early, it could short circuit a student’s chance to truly understand the concept of simplifying square roots.

    Thanks for sharing an interesting method!

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  10. Joyce

    I can see where this method would have some gtreat advantages. I teach middle school math, and needless to say, most students at that age have less than neat handwriting. When drawing the traditional prime factor tree, they “lose” a number once in a while. Some of the students are pursuing a math related career, and need to fully understand the concept of the traditional tree, but others need a “survival technique” and I think this might do it. We get too concerned sometimes that they understand every concept as deeply as we understand it. When they just don’t get it, and don’t have a survival technique, they become totally turned off to math. Then they won’t get anything. Show it to them both ways, and toss that life raft to those drowning in math problems!